Instanton bundles vs Ulrich bundles on projective spaces
Ulrich bundles have been an object of interest in algebra geometry in recent times, being the geometric counterpart of a kind of Maximally Cohen-Macaulay modules with extremal properties. In [J. Am. Math. Soc. 16, No. 3, 537--575, appendix 576--579 (2003; Zbl 1069.14019)], \textit{D. Eisenbud} and \textit{F.-O. Schreyer} raised the question whether any projective variety supports Ulrich sheaves. This paper contributes to this question for the case of the Veronese threefold \(V=(\mathbb{P}^3,\mathcal{O}_{\mathbb{P}^3}(d))\). In order to do it, the authors relate Ulrich bundles with the well-studied family of instanton bundles on \(\mathbb{P}^3\). In particular, they show there exists a correspondence between Ulrich bundles on \(V\) and instanton bundles with natural cohomology and quantum number \(r(d^2-1)/6\). This relation prompts them to conjecture that an Ulrich bundle on \(V\) of rank \(r\) exists if and only if \(r(d^2-1)\) is divisible by \(6\). They solve the conjecture for many values of \((d,r)\).
scientific article; zbMATH DE number 7352562